GROUP OF ORDER 64 # 96

GROUP # 96

The MAGMA library number for G is 89

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.3^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.5 * G.6, G.4^G.1 = G.4 * G.6, G.4^G.2 = G.4 * G.5

The center of G is abelian of type [ 2, 4 ]
The orders of the terms of the lower central series are [ 64, 4, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 64 ]
The order of the Frattini subgroup is 8
The exponent of G is 8
G has 3 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8, 8 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 16, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64, 64 ]

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u, t ] in degrees [ 1, 1, 1, 2, 2, 3, 4 ] by the ideal generated by the relations
[ z^2, zy + zx, y^2x + zx^2 + yx^2, zw, zx^3 + yx^3 + x^4 + w^2, yxw + yxv + x^2v + yu + xu, zxv + yxv + x^2v + zu + yu + xu, yx^2v + x^3v + xwv + zxu + yxu + x^2u + wu, x^4v + x^2v^2 + w^2*v + u^2 ]

The Hilbert series for the cohomology ring is -1/(t^3 - 3*t^2 + 3*t - 1)
Its denominator factors as ( t - 1 )^3

The Krull dimension of the cohomology ring is 3
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ v, t, y^2 + y*x + x^2 ]

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.2 * G.4, G.5, G.5 * G.6 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z, z, 0, y^2, z^2y + zy^2, z^2y^2 + y^4 + z^2x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y + x, w ]

Subgroup E # 2
Generated by [ G.5, G.5 * G.6, G.3 * G.4 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, z, z^2, zy + y^2, z^2y + zy^2, z^2y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y, x^2 + w, x*v + u ]

Subgroup E # 3
Generated by [ G.2 * G.5, G.5, G.5 * G.6 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z, 0, 0, zy + y^2, 0, z^2y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, x, w, u ]

The nilradical of the cohomology of G is generated by
[ z, y*x + x^2 + w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2, G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z, y, x, y^2 + x + w, y^3 + yx + zw + yw + zv, y^2*v + w^2 + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y ]

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4 * G.6, G.2 * G.3 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, z, y^2 + x, y^2 + x + w, zw + zv, y^2*v + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + x ]

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.4, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ z, 0, y, x, w, yw + zv, y^2*v + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y ]

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, 0, y^2 + x, w, zw + zv, y^2*v + w^2 + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ x ]

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.6, G.2 * G.4, G.2 * G.3 * G.6, G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ 0, z + y + x + w, z + w, zy + y^2 + zx + yx, y^2 + yw + xw, zy^2 + y^3 + z^2w + yw^2, z^2yx + z^2yw + y^3w + y^2w^2 + yxw^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z ]

Maximal Subgroup H # 6

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z + y + x, y + x, zy, y^2, zy^2 + y^2x + yx^2 + zw, y^2x^2 + y^2w + x^2w + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y + x ]

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.2 * G.3 * G.4 * G.5, G.1 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + x, z + x, zy + y^2, zy, z^2y + x^3 + w, z^3x + z^2yx + zyx^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y + x ]

The essential cohomology of G is zero

CHECKS

paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true
bigflag = true
restrictflag = true

THE COMPUTATION IS COMPLETE

GROUP # 96

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.3^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.5 * G.6, G.4^G.1 = G.4 * G.6, G.4^G.2 = G.4 * G.5

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1 Generated by [ G.2 * G.4, G.5, G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to E are [ 0, z, z, 0, y^2, z^2*y + z*y^2, z^2*y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by [ z, y + x, w ]

Subgroup E # 2 Generated by [ G.5, G.5 * G.6, G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to E are [ 0, 0, z, z^2, z*y + y^2, z^2*y + z*y^2, z^2*y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by [ z, y, x^2 + w, x*v + u ]

Subgroup E # 3 Generated by [ G.2 * G.5, G.5, G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to E are [ 0, z, 0, 0, z*y + y^2, 0, z^2*y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by [ z, x, w, u ]

The nilradical of the cohomology of G is generated by [ z, y*x + x^2 + w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2, G.4 ]

The images of the generators of the cohomology of G restricted to H are [ z, z, y, x, y^2 + x + w, y^3 + y*x + z*w + y*w + z*v, y^2*v + w^2 + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ z + y ]

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4 * G.6, G.2 * G.3 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are [ z, y, z, y^2 + x, y^2 + x + w, z*w + z*v, y^2*v + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ z + x ]

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.4, G.1 ]

The images of the generators of the cohomology of G restricted to H are [ z, 0, y, x, w, y*w + z*v, y^2*v + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ y ]

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are [ z, y, 0, y^2 + x, w, z*w + z*v, y^2*v + w^2 + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ x ]

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.6, G.2 * G.4, G.2 * G.3 * G.6, G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are [ 0, z + y + x + w, z + w, z*y + y^2 + z*x + y*x, y^2 + y*w + x*w, z*y^2 + y^3 + z^2*w + y*w^2, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ z ]

Maximal Subgroup H # 6

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations [ z^2 ]

The images of the generators of the cohomology of G restricted to H are [ z, z + y + x, y + x, z*y, y^2, z*y^2 + y^2*x + y*x^2 + z*w, y^2*x^2 + y^2*w + x^2*w + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ z + y + x ]

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.2 * G.3 * G.4 * G.5, G.1 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are [ z + y, z + x, z + x, z*y + y^2, z*y, z^2*y + x^3 + w, z^3*x + z^2*y*x + z*y*x^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by [ y + x ]

The essential cohomology of G is zero

CHECKS

THE COMPUTATION IS COMPLETE

Subgroup E # 1
Generated by [ G.2 * G.4, G.5, G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z, z, 0, y^2, z^2y + zy^2, z^2y^2 + y^4 + z^2x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y + x, w ]

Subgroup E # 2
Generated by [ G.5, G.5 * G.6, G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, z, z^2, zy + y^2, z^2y + zy^2, z^2y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y, x^2 + w, x*v + u ]

Subgroup E # 3
Generated by [ G.2 * G.5, G.5, G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z, 0, 0, zy + y^2, 0, z^2y^2 + y^4 + z^2*x^2 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, x, w, u ]

The nilradical of the cohomology of G is generated by
[ z, y*x + x^2 + w ]

The images of the generators of the cohomology of G restricted to H are
[ z, z, y, x, y^2 + x + w, y^3 + yx + zw + yw + zv, y^2*v + w^2 + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, z, y^2 + x, y^2 + x + w, zw + zv, y^2*v + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + x ]

The images of the generators of the cohomology of G restricted to H are
[ z, 0, y, x, w, yw + zv, y^2*v + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, 0, y^2 + x, w, zw + zv, y^2*v + w^2 + v^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ x ]

The images of the generators of the cohomology of G restricted to H are
[ 0, z + y + x + w, z + w, zy + y^2 + zx + yx, y^2 + yw + xw, zy^2 + y^3 + z^2w + yw^2, z^2yx + z^2yw + y^3w + y^2w^2 + yxw^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z + y + x, y + x, zy, y^2, zy^2 + y^2x + yx^2 + zw, y^2x^2 + y^2w + x^2w + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y + x ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + x, z + x, zy + y^2, zy, z^2y + x^3 + w, z^3x + z^2yx + zyx^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y + x ]